The Gromov-Wasserstein distance and Sturm's distance both satisfy metric properties, up to an invariance to isometry.
One defines the equivalence relation \(\sim \)  between mm-spaces if there exists an isometry \(\psi :\rightarrow \)  such that \(d_(x,x^{\prime })=d_(\psi (x), \psi (x^{\prime }))\)  and \(\psi _\sharp =\) .
Note that \(\psi \)  preserves both the metric isometry and the measures, and that \((,d_,)\sim ((),d_,)\) .
Thus mm-spaces are only compared on the measures' supports (not necessarily on the entire spaces \(,\) ).
Both distances are zero if and only if \(\sim \) , assuming \((,)\)  are endowed with a complete metric.
Thus Sturm's and GW distances are able to distinguish mm-spaces up to this equivalence relation.
However, the two distances [1]}, [2]} are not equivalent.
For instance, there are differences in topological properties.
Sturm's distance is complete over the set of mm-spaces endowed with a complete metric, while \(\)  is not.
Gromov-Wasserstein (and Sturm's) distance is a complete metric if one considers instead the set of mm-spaces endowed with a pseudo-metric (i.e. \(d_\)  or \(d_\)  are not necessarily definite, but still satisfy \(d_(x,x)=0\)  for any \(x\in \) ) [2]}.
